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The derivative can be found by using the chain rule. i.e. let g(x) = |sin(x)|, so f(x)=ln(g(x)), hence df/dx = df/dg * dg/dx

df/dg = 1/g, dg/dx = |cos(x)| so df/dx = |cos(x)|/|sin(x)|

For th...

Implicit differentiation involves differentiating expressions for which it is difficult to rearrange into y in terms of x. It is important to note which variable is being differentiated by the other as, f...

Ratios are a way of comparing the amount of two things in a whole. In this case we are comparing the number of men and women in a football match, so, the total number of people at the match is equal to th...

For part i) we use the basic method of differentiation by considering each term individually. The first term, x³ goes to 3x² by multiplying the original term by the original power, b...

Rearrange the terms and group: Numbers: (3*1) =3, a variables: a * (a^3) --> (1+3=4) Therefore (a^4); b variables: (b^2)*b --> (2+1=3) Therefore (b^3); Regroup: 3(a^4)(b^3)